Why isn't the texture in rendered view transparent but the material view is
You're using the cycles render so you have to open up the "Node Editor" window and add in what is called a "Transparent BSDF" and a "Mix Shader".


Simply connect the transparent bsdf to the mix shader and then
also connect the "diffuse bsdf" (which should already be there) to the same mix shader.
Connect the mix shader to the "Material Output" (already there). 
and finally connect the alpha channel of the image texture node (already there) into the Fac of the mix shader.


and thats it, super easy. In the event you're using principled bsdf node instead of diffuse bsdf then its the same steps but with the principled node instead of the diffuse node.

More thorough answer here - https://www.youtube.com/watch?v=AdrFMEgNgF4 starting at about 9:40